Subgroups — C₃×#3294 #85158

Order 384.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#480	C₂	1	2	yes	maximal, complemented
5	C₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#466	C₂	1	2	yes	maximal, complemented
7	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#447	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#102	C₂	1	2	yes	maximal, complemented
12	#3294	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	—	2	1	no	—
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	2	yes	—
16	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×D₈	C₂²	1	2	yes	complemented
18	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
19	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×#127	C₂²	1	1	yes	characteristic
21	C₃×#123	—	2	2	no	—
22	C₃×C₂²⋊Q₈	—	2	1	no	—
23	C₃×#121	—	2	1	no	—
24	C₃×#121	C₂²	1	2	yes	—
25	C₃×#121	C₂²	1	2	yes	complemented
26	C₃×#120	—	2	2	no	—
27	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
31	C₃×#107	C₂²	1	1	yes	characteristic, complemented
32	C₃×#107	C₂²	1	2	yes	complemented
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃×#102	C₂²	1	1	yes	characteristic
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	complemented
37	C₃×#102	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	1	yes	characteristic, complemented
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
42	#547	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄²	C₂×C₃	1	1	yes	characteristic, complemented
44	#486	C₂×C₃	1	1	yes	characteristic, complemented
45	#480	C₂×C₃	1	2	yes	complemented
46	#474	C₂×C₃	1	1	yes	characteristic, complemented
47	#466	C₂×C₃	1	2	yes	complemented
48	#459	C₂×C₃	1	1	yes	characteristic, complemented
49	#447	C₂×C₃	1	2	yes	complemented
50	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
51	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×#102	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃	—	2	2	no	—
54	C₃×#41	—	4	4	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×D₄	—	2	4	no	—
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	4	2	no	—
63	C₂×C₃×D₄	—	4	4	no	—
64	C₂×C₃×D₄	D₄	1	1	yes	characteristic
65	C₂×C₃×D₄	C₂³	1	2	yes	—
66	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄	—	2	2	no	—
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	C₂³	1	2	yes	—
72	C₃×D₈	—	2	2	no	—
73	C₃×D₈	—	4	2	no	—
74	C₂×C₃×C₈	—	4	2	no	—
75	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂³	1	2	yes	—
78	C₃×C₄⋊C₄	—	2	2	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₄⋊C₄	C₂³	1	2	yes	—
83	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₂²⋊C₄	—	2	2	no	—
87	C₃×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₂²⋊C₄	D₄	1	2	yes	—
89	C₃×C₄²	C₂³	1	1	yes	characteristic
90	C₂×#41	—	2	1	no	—
91	C₂²×D₄	—	2	1	no	—
92	C₂²×D₄	C₂²×C₃	1	2	yes	—
93	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
94	C₂×D₈	C₂²×C₃	1	2	yes	complemented
95	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
96	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
97	#127	C₂²×C₃	1	1	yes	characteristic
98	#123	—	2	2	no	—
99	C₂²⋊Q₈	—	2	1	no	—
100	#121	—	2	1	no	—
101	#121	C₂²×C₃	1	2	yes	—
102	#121	C₂²×C₃	1	2	yes	complemented
103	#120	—	2	2	no	—
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
105	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
106	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
107	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
108	#107	C₂²×C₃	1	1	yes	characteristic, complemented
109	#107	C₂²×C₃	1	2	yes	complemented
110	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
111	#102	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	2	yes	complemented
114	#102	C₂²×C₃	1	2	yes	complemented
115	#102	C₂²×C₃	1	1	yes	characteristic, complemented
116	#102	C₂²×C₃	1	2	yes	complemented
117	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
118	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	4	4	no	—
123	C₂³×C₃	—	4	4	no	—
124	C₂³×C₃	—	8	2	no	—
125	C₂³×C₃	C₂×D₄	1	2	yes	—
126	C₃×Q₈	—	4	2	no	—
127	C₃×D₄	—	2	1	no	—
128	C₃×D₄	—	2	2	no	—
129	C₃×D₄	—	2	1	no	—
130	C₃×D₄	—	2	1	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	4	4	no	—
133	C₃×D₄	—	4	4	no	—
134	C₃×D₄	—	4	4	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	—	8	1	no	—
137	C₃×D₄	D₈	1	4	yes	complemented
138	C₃×D₄	#41	1	2	yes	complemented
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	2	2	no	—
154	C₂⁴	—	2	2	no	—
155	#41	—	4	4	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	C₃×D₄	1	1	yes	characteristic
166	C₂×D₄	C₂³×C₃	1	2	yes	—
167	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂³×C₃	1	2	yes	—
173	D₈	—	2	2	no	—
174	D₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₃	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
185	C₂²⋊C₄	—	2	1	no	—
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	C₃×D₄	1	2	yes	—
190	C₄²	C₂³×C₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	4	1	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	8	8	no	—
203	C₂²×C₃	C₂×D₈	1	4	yes	—
204	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	4	2	no	—
210	C₃×C₄	C₂×D₈	1	2	yes	—
211	C₃×C₄	C₂×#41	1	1	yes	characteristic
212	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	2	no	—
219	C₂³	C₂×C₃×D₄	1	2	yes	—
220	Q₈	—	4	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	2	no	—
223	D₄	—	2	1	no	—
224	D₄	—	2	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	8	1	no	—
231	D₄	C₃×D₈	1	4	yes	complemented
232	D₄	C₃×#41	1	2	yes	complemented
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
243	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	2	no	—
248	C₂×C₃	—	2	4	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	—	8	1	no	—
251	C₂×C₃	—	8	2	no	—
252	C₂×C₃	#546	1	1	yes	characteristic
253	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
254	C₂×C₃	#577	1	1	yes	characteristic
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	8	8	no	—
267	C₂²	C₂×C₃×D₈	1	4	yes	—
268	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	C₂×C₃×D₈	1	2	yes	—
275	C₄	C₂×C₃×#41	1	1	yes	characteristic
276	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
277	C₃	#3294	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	8	2	no	—
281	C₂	—	8	1	no	—
282	C₂	C₃×#546	1	1	yes	characteristic
283	C₂	C₂²×C₃×D₈	1	1	yes	characteristic
284	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#547	C₂	semidirect
2. C₃×D₄²	C₂	semidirect
3. C₃×#486	C₂	semidirect
4. C₃×#480	C₂	semidirect
5. C₃×#474	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#459	C₂	semidirect
8. C₃×#447	C₂	semidirect
9. C₃×C₄×D₈	C₂	semidirect
10. C₃×C₈×D₄	C₂	semidirect
11. C₂×C₃×#102	C₂	semidirect
12. #3294	C₃	direct (N × H)
17. C₂×C₃×D₈	C₂²	semidirect
18. C₂²×C₃×C₈	C₂²	semidirect
19. C₃×C₄⋊Q₈	C₂²	semidirect
25. C₃×#121	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
30. C₂×C₃×C₄⋊C₄	C₂²	semidirect
31. C₃×#107	C₂²	semidirect
32. C₃×#107	C₂²	semidirect
33. C₃×C₄⋊C₈	C₂²	semidirect
36. C₃×#102	C₂²	semidirect
37. C₃×#102	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
42. #547	C₂×C₃	semidirect
43. D₄²	C₂×C₃	semidirect
44. #486	C₂×C₃	semidirect
45. #480	C₂×C₃	semidirect
46. #474	C₂×C₃	semidirect
47. #466	C₂×C₃	semidirect
48. #459	C₂×C₃	semidirect
49. #447	C₂×C₃	semidirect
50. C₄×D₈	C₂×C₃	semidirect
51. C₈×D₄	C₂×C₃	semidirect
52. C₂×#102	C₂×C₃	semidirect
94. C₂×D₈	C₂²×C₃	semidirect
95. C₂²×C₈	C₂²×C₃	semidirect
96. C₄⋊Q₈	C₂²×C₃	semidirect
102. #121	C₂²×C₃	semidirect
106. C₄×D₄	C₂²×C₃	semidirect
107. C₂×C₄⋊C₄	C₂²×C₃	semidirect
108. #107	C₂²×C₃	semidirect
109. #107	C₂²×C₃	semidirect
110. C₄⋊C₈	C₂²×C₃	semidirect
113. #102	C₂²×C₃	semidirect
114. #102	C₂²×C₃	semidirect
115. #102	C₂²×C₃	semidirect
116. #102	C₂²×C₃	semidirect
117. C₂²⋊C₈	C₂²×C₃	semidirect
118. C₄×C₈	C₂²×C₃	semidirect
137. C₃×D₄	D₈	semidirect
138. C₃×D₄	#41	semidirect
231. D₄	C₃×D₈	semidirect
232. D₄	C₃×#41	semidirect
277. C₃	#3294	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#547	2
C₃×D₄²	2
C₃×#486	2
C₃×#480	2
C₃×#474	2
C₃×#466	2
C₃×#459	2
C₃×#447	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
C₂×C₃×#102	2
#3294	3

Subgroups — C3×#3294 #85158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3294 #85158